3 fixing grammar mistake, clarifying example

With regards to 2 and 3, if you have a MAD family of size $\kappa$ on $cf(\mu)$ then you have a MAD family of size $\kappa$ on $\mu$. (So, for example the there would be a 'degenerate families family' of the form you mention in 2 would occur on $\aleph_\omega$ if there is a MAD family of size $\aleph_1$ on $\omega$).

Let me give the argument for $\mu=\aleph_\omega$. Suppose $\mathcal{A}$ is a MAD family on $\omega$. For each $A\subseteq\omega$, let $B(A)\subseteq\aleph_\omega$ be the union of the intervals $I_n=[\aleph_n,\aleph_{n+1})$ such that $n\in A$. Let $\mathcal{B}$ be all the $B(A)$, for $A\in\mathcal{A}$. We claim that $\mathcal{B}$ is MAD.

The only interesting thing to check is the maximality. Suppose that $C\subseteq\aleph_\omega$ has cardinality $\aleph_\omega$. Let $n_0 < n_1 < n_2 < \ldots$ be a sequence such that $|I_{n_{k+1}}\cap C|\geq\aleph_{{n_k}+2}$. (The +2 is there to make sure the $n_{k+1}$ we find must be bigger than $n_k$). Letting $X=\{n_k:k\in\omega\}$, there is $Y\in\mathcal{A}$ with $X\cap Y$ infinite. Then $B(Y)\cap C$ has cardinality $\aleph_\omega$.

As for general references to MAD families on singular cardinals, it looks like you should check out Erdos, Hechler's "On Maximal Almost-Disjoint Families over Singular Cardinals" http://www.renyi.hu/~p_erdos/1975-21.pdf and the more recent Kojman, Kubis and Shelah "On Two Problems of Erdos and Hechler: New Methods in Singular MADness" http://arxiv.org/abs/math/0406441

2 added 374 characters in body

With regards to 2 and 3, if you have a MAD family of size $\kappa$ on $cf(\mu)$ then you have a MAD family of size $\kappa$ on $\mu$. (So, for example the degenerate families you mention in 2 would occur if there is a MAD family of size $\aleph_1$ on $\omega$).

Let me give the argument for $\mu=\aleph_\omega$. Suppose $\mathcal{A}$ is a MAD family on $\omega$. For each $A\subseteq\omega$, let $B(A)\subseteq\aleph_\omega$ be the union of the intervals $I_n=[\aleph_n,\aleph_{n+1})$ such that $n\in A$. Let $\mathcal{B}$ be all the $B(A)$, for $A\in\mathcal{A}$. We claim that $\mathcal{B}$ is MAD.

The only interesting thing to check is the maximality. Suppose that $C\subseteq\aleph_\omega$ has cardinality $\aleph_\omega$. Let $n_0 < n_1 < n_2 < \ldots$ be a sequence such that $|I_{n_{k+1}}\cap C|\geq\aleph_{{n_k}+2}$. (The +2 is there to make sure the $n_{k+1}$ we find must be bigger than $n_k$). Letting $X=\{n_k:k\in\omega\}$, there is $Y\in\mathcal{A}$ with $X\cap Y$ infinite. Then $B(Y)\cap C$ has cardinality $\aleph_\omega$.

As for general references to MAD families on singular cardinals, it looks like you should check out Erdos, Hechler's "On Maximal Almost-Disjoint Families over Singular Cardinals" http://www.renyi.hu/~p_erdos/1975-21.pdf and the more recent Kojman, Kubis and Shelah "On Two Problems of Erdos and Hechler: New Methods in Singular MADness" http://arxiv.org/abs/math/0406441

1

With regards to 2 and 3, if you have a MAD family of size $\kappa$ on $cf(\mu)$ then you have a MAD family of size $\kappa$ on $\mu$. (So, for example the degenerate families you mention in 2 would occur if there is a MAD family of size $\aleph_1$ on $\omega$).

Let me give the argument for $\mu=\aleph_\omega$. Suppose $\mathcal{A}$ is a MAD family on $\omega$. For each $A\subseteq\omega$, let $B(A)\subseteq\aleph_\omega$ be the union of the intervals $I_n=[\aleph_n,\aleph_{n+1})$ such that $n\in A$. Let $\mathcal{B}$ be all the $B(A)$, for $A\in\mathcal{A}$. We claim that $\mathcal{B}$ is MAD.

The only interesting thing to check is the maximality. Suppose that $C\subseteq\aleph_\omega$ has cardinality $\aleph_\omega$. Let $n_0 < n_1 < n_2 < \ldots$ be a sequence such that $|I_{n_{k+1}}\cap C|\geq\aleph_{{n_k}+2}$. (The +2 is there to make sure the $n_{k+1}$ we find must be bigger than $n_k$). Letting $X=\{n_k:k\in\omega\}$, there is $Y\in\mathcal{A}$ with $X\cap Y$ infinite. Then $B(Y)\cap C$ has cardinality $\aleph_\omega$.